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Complex Brjuno functions

Authors: Stefano Marmi, Pierre Moussa and Jean-Christophe Yoccoz
Journal: J. Amer. Math. Soc. 14 (2001), 783-841
MSC (2000): Primary 37F50, 11A55, 32A40; Secondary 37F25, 46F15, 20G99
Published electronically: May 30, 2001
MathSciNet review: 1839917
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The Brjuno function arises naturally in the study of analytic small divisors problems in one dimension. It belongs to $\hbox {BMO}({\mathbb{T}}^{1})$ and it is stable under Hölder perturbations. It is related to the size of Siegel disks by various rigorous and conjectural results.

In this work we show how to extend the Brjuno function to a holomorphic function on ${\mathbb{H}}/{\mathbb{Z}}$, the complex Brjuno function. This has an explicit expression in terms of a series of transformed dilogarithms under the action of the modular group. The extension is obtained using a complex analogue of the continued fraction expansion of a real number. Since our method is based on the use of hyperfunctions, it applies to less regular functions than the Brjuno function and it is quite general.

We prove that the harmonic conjugate of the Brjuno function is bounded. Its trace on ${\mathbb{R}}/{\mathbb{Z}}$ is continuous at all irrational points and has a jump of $\pi /q$ at each rational point $p/q\in {\mathbb{Q}}$.

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  • [BG1] A. Berretti and G. Gentile, Scaling properties for the radius of convergence of the Lindstedt series: the standard map, J. Math. Pures Appl. (9) 78 (1999), no. 2, 159-176. MR 2000c:37054
  • [BG2] A. Berretti and G. Gentile, Bryuno function and the standard map, University of Roma (Italy), Preprint (1998).
  • [BPV] N. Buric, I. Percival and F. Vivaldi, Critical function and modular smoothing, Nonlinearity 3 (1990), 21-37. MR 90m:58062
  • [Br1] A. D. Brjuno, Analytical form of differential equations, Trans. Moscow Math. Soc. 25 (1971), 131-288. MR 50:9476
  • [Br2] A. D. Brjuno, Analytical form of differential equations, Trans. Moscow Math. Soc. 26 (1972), 199-239. MR 50:9476
  • [Da1] A. M. Davie, The critical function for the semistandard map, Nonlinearity 7 (1994), 219-229. MR 95f:58067
  • [Da2] A. M. Davie, Renormalisation for analytic area-preserving maps, University of Edinburgh preprint (1995).
  • [Du] P. L. Duren, Theory of $H^{p}$ Spaces, Academic Press, New York, 1970. MR 42:3552
  • [Ga] J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. MR 83g:30037
  • [GCRF] J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, vol. 116, North Holland Mathematical Studies, Amsterdam, 1985. MR 87d:42023
  • [HW] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (Fifth Edition), Oxford Science Publications, 1979. MR 81i:10002
  • [H] L. Hörmander, The Analysis of Linear Partial Differential Operators I, vol. 256, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983. MR 85g:35002a
  • [Ja] N. Jacobson, Basic Algebra I and II, Freeman, San Francisco, 1980. MR 50:9457; MR 81g:00001
  • [KH] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, vol. 54, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1995. MR 96c:58055
  • [Le] L. Lewin, Polylogarithms and Associated Functions, Elsevier North-Holland, New York, 1981. MR 83b:33019
  • [Ma] S. Marmi, Critical functions for complex analytic maps, J. Phys. A: Math. Gen. 23 (1990), 3447-3474. MR 92b:58199
  • [MMY] S. Marmi, P. Moussa and J.-C. Yoccoz, The Brjuno functions and their regularity properties, Commun. Math. Phys. 186 (1997), 265-293. MR 98e:58137
  • [O] J. Oesterlé, Polylogarithmes, Séminaire Bourbaki n. 762, Astérisque 216 (1993), 49-67. MR 94m:11135
  • [Ri] E. Risler, Linéarisation des perturbations holomorphes des rotations et applications, Mémoires Soc. Math. France 77 (1999). CMP 2000:17
  • [S] C. L. Siegel, Iteration of analytic functions, Annals of Mathematics 43 (1942), 807-812. MR 4:76c
  • [Se] J.-P. Serre, Cohomologie Galoisienne, vol. 5, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1973. MR 53:8030
  • [St] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. MR 44:7280
  • [Yo1] J.-C. Yoccoz, Théorème de Siegel, nombres de Bruno et polynômes quadratiques, Astérisque 231 (1995), 3-88. MR 96m:58214
  • [Yo2] J.-C. Yoccoz, An introduction to small divisors problems, From Number Theory to Physics (M. Waldschmidt, P. Moussa, J.-M. Luck and C. Itzykson, eds.), Springer-Verlag, 1992, pp. 659-679. MR 94h:58148
  • [Yo3] J.-C. Yoccoz, Analytic linearisation of analytic circle diffeomorphisms, in preparation.

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Additional Information

Stefano Marmi
Affiliation: Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, Loc. Rizzi, I-33100 Udine, Italy
Address at time of publication: Scuola Normale Superiore, Classe di Scienze, Piazza dei Cavalieri 7, I-56126 Pisa, Italy

Pierre Moussa
Affiliation: Service de Physique Théorique, CEA/Saclay, 91191 Gif-Sur-Yvette, France

Jean-Christophe Yoccoz
Affiliation: Collège de France, 3 Rue d’Ulm, F-75005 Paris, France, and Université de Paris-Sud, Mathématiques, Batiment 425, F-91405 Orsay, France

Keywords: Small divisors, continued fractions, Bruno functions, complex boundary behaviour, renormalisation, hyperfunctions, modular group, dilogarithm
Received by editor(s): February 16, 2000
Published electronically: May 30, 2001
Additional Notes: This work began during a visit of the first author at the S.Ph.T.–CEA/Saclay and at the Department of Mathematics of Orsay during the academic year 1993–1994. This research has been supported by the CNR, CNRS, INFN, MURST and an EEC grant
Dedicated: This paper is dedicated to Michael R. Herman
Article copyright: © Copyright 2001 American Mathematical Society